Average Error: 30.5 → 5.5
Time: 8.3s
Precision: 64
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 2.3467556829869274:\\ \;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{1}{\frac{\varepsilon}{{x}^{3}}}, 1 - 0.5 \cdot {x}^{2}\right)} \cdot \left({1}^{\frac{1}{3}} - 0.166666666666666657 \cdot \left({x}^{2} \cdot {1}^{\frac{1}{3}}\right)\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1 + \frac{1}{\varepsilon}}{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}\right)\right)}\\ \end{array}\]
\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}
↓
\begin{array}{l}
\mathbf{if}\;x \le 2.3467556829869274:\\
\;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{1}{\frac{\varepsilon}{{x}^{3}}}, 1 - 0.5 \cdot {x}^{2}\right)} \cdot \left({1}^{\frac{1}{3}} - 0.166666666666666657 \cdot \left({x}^{2} \cdot {1}^{\frac{1}{3}}\right)\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}\\

\mathbf{else}:\\
\;\;\;\;e^{\log \left(\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1 + \frac{1}{\varepsilon}}{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}\right)\right)}\\

\end{array}
double f(double x, double eps) {
        double r55760 = 1.0;
        double r55761 = eps;
        double r55762 = r55760 / r55761;
        double r55763 = r55760 + r55762;
        double r55764 = r55760 - r55761;
        double r55765 = x;
        double r55766 = r55764 * r55765;
        double r55767 = -r55766;
        double r55768 = exp(r55767);
        double r55769 = r55763 * r55768;
        double r55770 = r55762 - r55760;
        double r55771 = r55760 + r55761;
        double r55772 = r55771 * r55765;
        double r55773 = -r55772;
        double r55774 = exp(r55773);
        double r55775 = r55770 * r55774;
        double r55776 = r55769 - r55775;
        double r55777 = 2.0;
        double r55778 = r55776 / r55777;
        return r55778;
}
↓
double f(double x, double eps) {
        double r55779 = x;
        double r55780 = 2.3467556829869274;
        bool r55781 = r55779 <= r55780;
        double r55782 = 1.3877787807814457e-17;
        double r55783 = 1.0;
        double r55784 = eps;
        double r55785 = 3.0;
        double r55786 = pow(r55779, r55785);
        double r55787 = r55784 / r55786;
        double r55788 = r55783 / r55787;
        double r55789 = 1.0;
        double r55790 = 0.5;
        double r55791 = 2.0;
        double r55792 = pow(r55779, r55791);
        double r55793 = r55790 * r55792;
        double r55794 = r55789 - r55793;
        double r55795 = fma(r55782, r55788, r55794);
        double r55796 = cbrt(r55795);
        double r55797 = 0.3333333333333333;
        double r55798 = pow(r55789, r55797);
        double r55799 = 0.16666666666666666;
        double r55800 = r55792 * r55798;
        double r55801 = r55799 * r55800;
        double r55802 = r55798 - r55801;
        double r55803 = r55796 * r55802;
        double r55804 = r55786 / r55784;
        double r55805 = fma(r55782, r55804, r55794);
        double r55806 = cbrt(r55805);
        double r55807 = r55803 * r55806;
        double r55808 = r55789 + r55784;
        double r55809 = r55808 * r55779;
        double r55810 = -r55809;
        double r55811 = exp(r55810);
        double r55812 = 2.0;
        double r55813 = r55811 / r55812;
        double r55814 = r55789 / r55784;
        double r55815 = r55789 - r55814;
        double r55816 = r55789 + r55814;
        double r55817 = r55789 - r55784;
        double r55818 = r55817 * r55779;
        double r55819 = exp(r55818);
        double r55820 = r55812 * r55819;
        double r55821 = r55816 / r55820;
        double r55822 = fma(r55813, r55815, r55821);
        double r55823 = log(r55822);
        double r55824 = exp(r55823);
        double r55825 = r55781 ? r55807 : r55824;
        return r55825;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 2.3467556829869274

    1. Initial program 39.5

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

    2. Simplified39.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1 + \frac{1}{\varepsilon}}{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}\right)}\]

    3. Taylor expanded around 0 7.2

      \[\leadsto \color{blue}{\left(1.38778 \cdot 10^{-17} \cdot \frac{{x}^{3}}{\varepsilon} + 1\right) - 0.5 \cdot {x}^{2}}\]

    4. Simplified7.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt7.2

      \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}}\]

    7. Taylor expanded around 0 7.0

      \[\leadsto \left(\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)} \cdot \color{blue}{\left({1}^{\frac{1}{3}} - 0.166666666666666657 \cdot \left({x}^{2} \cdot {1}^{\frac{1}{3}}\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}\]
    8. Using strategy rm

    9. Applied clear-num7.0

      \[\leadsto \left(\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \color{blue}{\frac{1}{\frac{\varepsilon}{{x}^{3}}}}, 1 - 0.5 \cdot {x}^{2}\right)} \cdot \left({1}^{\frac{1}{3}} - 0.166666666666666657 \cdot \left({x}^{2} \cdot {1}^{\frac{1}{3}}\right)\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}\]

    if 2.3467556829869274 < x

    1. Initial program 0.6

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

    2. Simplified0.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1 + \frac{1}{\varepsilon}}{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}\right)}\]
    3. Using strategy rm

    4. Applied add-exp-log0.6

      \[\leadsto \color{blue}{e^{\log \left(\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1 + \frac{1}{\varepsilon}}{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}\right)\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.3467556829869274:\\ \;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{1}{\frac{\varepsilon}{{x}^{3}}}, 1 - 0.5 \cdot {x}^{2}\right)} \cdot \left({1}^{\frac{1}{3}} - 0.166666666666666657 \cdot \left({x}^{2} \cdot {1}^{\frac{1}{3}}\right)\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(1.38778 \cdot 10^{-17}, \frac{{x}^{3}}{\varepsilon}, 1 - 0.5 \cdot {x}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\mathsf{fma}\left(\frac{e^{-\left(1 + \varepsilon\right) \cdot x}}{2}, 1 - \frac{1}{\varepsilon}, \frac{1 + \frac{1}{\varepsilon}}{2 \cdot e^{\left(1 - \varepsilon\right) \cdot x}}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))